To begin with this section, we now recall Schrödinger-type representations
from Chapters 1 and 6 in the book [1]}. The equivalent
realization of Schrödinger-type representations \(\lbrace \pi _{\lambda }\rbrace _{\lambda \in \mathbb {R}^*}\)  of \(\mathbf {H}_n\)  on \(L^2(\mathbb {R}^n)\)  can be represented by
\(\pi _{\lambda }(x,y,\tau )\varphi (w):=\mathrm {e}^{i\lambda (\tau +\frac{1}{2}x\cdot y)+i\,\mathrm {sign}(\lambda )\sqrt{|\lambda |}\,y\cdot w}\varphi (w+\sqrt{|\lambda |}x),\) 
